D32.3 Second-Order Reaction
For the generic reaction “A ⟶ products”, the integrated rate law is this:
![\displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} \dfrac{d[\text{A}]}{[\text{A}]^m} = -kt}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-223fa25d0949cbede50ff4a6d6282ad9_l3.png "Rendered by QuickLaTeX.com")
When the reaction is second order with respect to [A], that is, when m = 2, this becomes:
![\begin{array}{rcl} \displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} \dfrac{d[\text{A}]}{[\text{A}]^2} &=& -kt \\[1.5em] -\dfrac{1}{[\text{A}]_t}-\left(-\dfrac{1}{[\text{A}]_0}\right) &=& -kt} \end{array}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-5b59ee027fd83940594ec211cc9fdd9a_l3.png "Rendered by QuickLaTeX.com")
This integrated rate law for a second-order reaction can be alternatively expressed as:
![\dfrac{1}{[\text{A}]_t} = kt\;+\;\dfrac{1}{[\text{A}]_0}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-6f92ca005437e2be6f9b43d4d8da5186_l3.png "Rendered by QuickLaTeX.com")
Exercise: Calculate Concentration from Time
The integrated rate law for a second-order reaction also has a standard linear equation format:
![\begin{array}{rcl} \dfrac{1}{[\text{A}]_t} &=& \; kt \; + \dfrac{1}{[\text{A}]_0} \\[1em] y &=& mx + \;\; b \end{array}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-7014b56e62672f2ed5ba62697b6bd3c9_l3.png "Rendered by QuickLaTeX.com")
Hence, if a reaction is second order in [A], a plot of ![\frac{1}{[\text{A}]_t}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-89e8266ce2a427b23c044589fcfcf10e_l3.png "Rendered by QuickLaTeX.com")
vs t should yield a straight line, where the slope equals k and the y-intercept is ![\frac{1}{[\text{A}]_0}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-f144d0a0f42541527b40048145b7ca14_l3.png "Rendered by QuickLaTeX.com")
. If the plot is not a straight line, then the reaction is not second order with respect to [A].
Activity: Order from Integrated Rate Law
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